With cedram.org   version française
Search for an article
Table of contents for this volume | Previous article | Next article
Robert C. Kirby
A general approach to transforming finite elements
SMAI-Journal of computational mathematics, 4 (2018), p. 197-224, doi: 10.5802/smai-jcm.33
Article PDF
Class. Math.: 65N30
Keywords: Finite element method, basis function, pull-back

Abstract

The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way. This paper gives a generalized approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward.

This approach, developed through a sequence of examples of increasing complexity, requires one to study the relationship between the function space and degrees of freedom, or nodes, on a generic cell and the transformation of the corresponding entities on a reference cell. When the function space is preserved under mapping, one must be able to express the pushed-forward finite element nodes as linear combinations of the reference element finite element nodes. The transpose of this linear transformation maps the pull-back of the reference element basis functions to the desired finite element basis functions. Generically, developing this transformation for elements such as Morley and Hermite involves completing the set of finite element nodes, although the process is simplified in concrete ways when the finite elements form affine- or affine-interpolation equivalent families such as Lagrange or Hermite. When the finite element function space is not preserved under the pull-back, such as in the case of the Bell element, one applies the theory to an enriched finite element with a larger (but preserved) function space and additional nodes.

Bibliography

[1] M. Ainsworth, G. Andriamaro & O. Davydov, “Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures”, SIAM Journal on Scientific Computing 33 (2011) no. 6, p. 3087-3109
[2] Martin S. Alnæs, Anders Logg, Kristian B. Ølgaard, Marie E. Rognes & Garth N. Wells, “Unified Form Language: A domain-specific language for weak formulations of partial differential equations”, ACM Transactions on Mathematical Software 40 (2014) no. 2
[3] J. H. Argyris, I. Fried & D. W. Scharpf, “The TUBA family of plate elements for the matrix displacement method”, Aeronautical Journal 72 (1968), p. 701-709
[4] W. Bangerth, R. Hartmann & G. Kanschat, “deal.II — a General Purpose Object Oriented Finite Element Library”, ACM Trans. Math. Softw. 33 (2007) no. 4 Article
[5] K. Bell, “A refined triangular plate bending finite element”, International Journal for Numerical Methods in Engineering 1 (1969) no. 1, p. 101-122
[6] M. Bernadou, Finite Element Methods for Thin Shell Problems, Wiley, 1996
[7] M. Bernadou & P.-L. George, MODULEF: une bibliothèque modulaire d’éléments finis, France. Inst. Nat. Rech. Inform. Autom., 1985
[8] P.B. Bochev, H. Carter Edwards, Robert C. Kirby, Kara Peterson & Denis Ridzal, “Solving PDEs with Intrepid”, Scientific Programming 20 (2012) no. 2, p. 151-180
[9] James H. Bramble & S. R. Hilbert, “Bounds for a class of linear functionals with applications to Hermite interpolation”, Numerische Mathematik 16 (1971) no. 4, p. 362-369
[10] Susanne C. Brenner & L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics 15, Springer, New York, 2008
[11] Franco Brezzi, Jim Douglas Jr. & L. Donatella Marini, “Two families of mixed finite elements for second order elliptic problems”, Numerische Mathematik 47 (1985) no. 2, p. 217-235
[12] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland, 1978
[13] Philippe G. Ciarlet & P. A. Raviart, “General Lagrange and Hermite interpolation in $\mathbb{R}^n$ with applications to finite element methods”, Archive for Rational Mechanics and Analysis 46 (1972) no. 3, p. 177-199
[14] Lisandro D. Dalcin, Rodrigo R. Paz, Pablo A. Kler & Alejandro Cosimo, “Parallel distributed computing using Python”, Advances in Water Resources 34 (2011) no. 9, p. 1124-1139, New Computational Methods and Software Tools Article
[15] Robert Dautray & Jacques-Louis Lions, Mathematical Analysis and Numerical Methods for Science and Technology 4: Integral Equations and Numerical Methods, Springer-Verlag, 2012
[16] Victor Domínguez & Francisco-Javier Sayas, “Algorithm 884: A simple Matlab implementation of the Argyris element”, ACM Transactions on Mathematical Software (TOMS) 35 (2008) no. 2
[17] Todd F. Dupont & L. Ridgway Scott, “Polynomial approximation of functions in Sobolev spaces”, Mathematics of Computation 34 (1980), p. 441-463
[18] G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei & R. L. Taylor, “Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity”, Computer Methods in Applied Mechanics and Engineering 191 (2002) no. 34, p. 3669-3750
[19] Michel Fortin & Franco Brezzi, Mixed and hybrid finite element methods, Springer, 1991
[20] Jan S. Hesthaven & Timothy Warburton, Nodal discontinuous Galerkin methods: Algorithms, analysis and applications, Springer Texts in Applied Mathematics 54, Springer-Verlag, 2008
[21] Miklós Homolya, Lawrence Mitchell, Fabio Luporini & David A. Ham, “TSFC: a structure-preserving form compiler”, https://arxiv.org/abs/1705.03667, 2017
[22] Stephen C. Jardin, “A triangular finite element with first-derivative continuity applied to fusion MHD applications”, Journal of Computational Physics 200 (2004) no. 1, p. 133-152
[23] Eric Jones, Travis Oliphant & Pearu Peterson, “SciPy: Open source scientific tools for Python”, http://www. scipy. org/, 2001
[24] Robert C. Kirby, “FIAT: A New Paradigm for Computing Finite Element Basis Functions”, ACM Trans. Math. Software 30 (2004), p. 502-516
[25] Robert C. Kirby, “Fast simplicial finite element algorithms using Bernstein polynomials”, Numerische Mathematik 117 (2011) no. 4, p. 631-652
[26] Robert C. Kirby & Anders Logg, “A Compiler for Variational Forms”, ACM Transactions on Mathematical Software 32 (2006) no. 3 Article
[27] Anders Logg, Kent-Andre Mardal, Garth N. Wells & , Automated Solution of Differential Equations by the Finite Element Method, Springer, 2012 Article
[28] Kevin R. Long, Robert C. Kirby & Bart van Bloemen Waanders, “Unified embedded parallel finite element computations via software-based Fréchet differentiation”, SIAM Journal on Scientific Computing 32 (2010) no. 6, p. 3323-3351
[29] Fabio Luporini, Ana Lucia Varbanescu, Florian Rathgeber, Gheorghe-Teodor Bercea, J. Ramanujam, David A. Ham & Paul H. J. Kelly, “COFFEE: an Optimizing Compiler for Finite Element Local Assembly”, https://arxiv.org/abs/1407.0904, 2014
[30] Kent-Andre Mardal, Xue-Cheng Tai & Ragnar Winther, “A robust finite element method for Darcy–Stokes flow”, SIAM Journal on Numerical Analysis 40 (2002) no. 5, p. 1605-1631
[31] L. S. D. Morley, “The constant-moment plate-bending element”, The Journal of Strain Analysis for Engineering Design 6 (1971) no. 1, p. 20-24
[32] Christophe Prud’homme, Vincent Chabannes, Vincent Doyeux, Mourad Ismail, Abdoulaye Samake & Gonçalo Pena, Feel++: A computational framework for Galerkin methods and advanced numerical methods, in ESAIM: Proceedings, EDP Sciences, 2012, p. 429-455
[33] Florian Rathgeber, David A. Ham, Lawrence Mitchell, Michael Lange, Fabio Luporini, Andrew T. T. McRae, Gheorghe-Teodor Bercea, Graham R. Markall & Paul H .J. Kelly, “Firedrake: automating the finite element method by composing abstractions”, ACM Transactions on Mathematical Software (TOMS) 43 (2016) no. 3
[34] Marie E. Rognes, Robert C. Kirby & Anders Logg, “Efficient Assembly of $H(\mathrm{div})$ and $H(\mathrm{curl})$ Conforming Finite Elements”, SIAM Journal on Scientific Computing 31 (2009) no. 6, p. 4130-4151 Article
[35] Garth N. Wells & Nguyen Tien Dung, “A $C^0$ discontinuous Galerkin formulation for Kirchhoff plates”, Computer Methods in Applied Mechanics and Engineering 196 (2007) no. 35, p. 3370-3380